Singular solutions of the turbulent boundary layer equations in the case of marginal separation as Re → ∞

We consider a nominally steady and two-dimensional turbulent boundary layer (BL) of uniform density along a flat surface under the action of an adverse pressure gradient. Classical analysis of the flow in the limit of large Reynolds number, Re → ∞, (a survey is found e.g. in [2]) predicts the well-known asymptotically small velocity defect holding within most of a strictly attached BL. However, as demonstrated in [1], a more general asymptotic approach accounting for a large velocity deficit and, in turn, the possibility of separation apparently requires the existence of a second perturbation parameter, denoted by α. The latter is (i) essentially independent of Re, (ii) serves as a measure for the BL slenderness and is (iii) in fact provided by the empirical constants entering any commonly employed Reynolds shear stress closure. Including α ≪ 1 in the theoretical considerations as first put forward by Melnik (1989), referred to in [1], has the important consequence that the BL thickness remains finite and of O(α) in the limit Re = 0, which will be considered here. To this end, let x, y, ψ, δ and l denote Cartesian coordinates parallel and normal to the wall, the stream function, the BL thickness and the mixing length, non-dimensional with global reference quantities, respectively. As shown in [1], appropriately scaled variables in the outer part of the BL are Y = y/α, Ψ = ψ/α, L = l/α, ∆ = δ/α which upon substitution into the set of Reynolds-averaged Navier–Stokes equations yield the leading-order problem